M a ssa c huset t s I nst it ut e of T ec hnol ogy
M at hem at ical M et hods
for M at er ials S cient ist s and E ngineer s
3.016 Fall 2005
W . C raig C arter
Department of Materials Science and Engineering
Massachusetts Institute of Technology
77 Massachusetts Ave.
Cambridge, MA 02139
P roblem Set 3: Due T hu. Oct. 12, B efore 5P M: email to the TA.
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Individual Exercise I3-1
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Computer Guide: problem 6.4, page 77
Kreyszig Mathematica
Individual Exercise I3-2
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Kreyszig Mathematica
Computer Guide: problem 7.10, page 87
Individual Exercise I3-3
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Computer Guide: problem 7.12, page 87
Kreyszig Mathematica
Group Exercise G3-1
An edge dislocation generates a stress field around it. A straight edge dislocation lying along the
z-axis will not generate forces in the z-direction and therefore its stress state can be represented
in two dimensions.
For an infinitely long edge dislocation with its extra lattice plane inserted the y > 0—z halfplane, the stress state is given by
� −y(3x2 +y2 ) x(x2 −y2 ) �
�
�
Gb
σxx σxy
(x2 +y 2 )2
(x2 +y 2 )2
=
2 −y 2 )
2 −y 2 )
y(x
x(x
σyx σyy
2π(1 − ν)
(x2 +y 2 )2
(x2 +y 2 )2
where b is the Burgers vector magnitude, and G and ν are the the shear modulus and Poisson’s
ratio for an isotropic elastic material.
1. Convert these stresses to a representation in terms of polar coordinates r, θ.
2. Calculate the hydrostatic pressure due to an edge as a function of position in the x, y-plane.
Plot some of the isobars (contours of constant pressure).
3. Calculate the rotation of the principal axis as a function of position and plot it.
4. The maximum shear stress is the σxy component rotated by π/4 from the principal axis
system—and the Mohr’s circle construction provides a simple way to calculate the maximum
shear stress in terms of the eigenvalues. Plot the maximum shear stress as a function of
position.
Hint: The principal coordinate system is a special coordinate system where the stress matrix is
a diagonal matrix. It is related by a rotation from the laboratory coordinate system in which the
problem is posed. The trace of a matrix does not change when the coordinate system is rotated.
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Group Exercise G3-2
The purpose of this problem is to calculate the entropy of a very small simple system exactly.
By solving this problem, I hope you will understand Boltzmann’s formula for entropy S(E) =
k log Ω(E) a little better and also understand why approximations are needed to calculate entropy
in larger or more complex systems. In Boltzmann’s formula, S is the entropy of the total system;
E is the energy of the total system; k is Boltzmann’s constant; Ω(E) is the number of states of
the system that have energy E.
Consider a system of three isolated hydrogen atoms.
Let the “zero of energy” be the ground state of the hydrogen atom, so that the energy of a
single hydrogen atom with its electron in state n is:
E(n) = Eo (1 −
1
)
n2
As you know, counting the quantum numbers for each energy state (electron spin s, angular
momentum l, etc.), there are two states for n = 1; eight states for n = 2; and 18 states for n = 3,
etc.
So that the problem can be done with a reasonably small amount of RAM, suppose that all
three electrons are either in n = 1 or n = 2 and no other states.
1. Calculate and illustrate the total entropy of this simple system of three non-interacting
hydrogen atoms;
2. Calculate S in multiples of k (e.g., plot S(E)/k vs. E).
3. Can you calculate S(E) for a system of three atoms if the quantum states are restricted to
n = 1, 2, 3? Five atoms?
4. Extra Credit: The assumption of a limited number of high-energy orbitals results in an an
unphysical result. Can you identify what is unphysical about your results?
Hint: dU = T dS − P dV for this simple system and the volume can be fixed.
Hint: One strategy is to enumerate all of the possible energies you can obtain by adding the all
the energies from two systems EA and EB ; call this EAB . For each energy in EAB , count number
of different ways each energy can be added up. Once you have established an algorithm for adding
two systems, you can add any number of systems by adding one at a time; i.e., combine EAB and
EC to get EABC .
Group Exercise G3-3
Recall from 3.012 that the time-independent Schrödinger equation is
Ĥφ = Eφ φ
where Ĥ is the Hamiltonian operator on the wavefunction φ and Eφ is the (scalar) eigenvalue for
the particular wavefunction φ. This is also an eigenvalue equation which is similar to the matrix–
eigenvector–eigenvalue systems we have been discussing, except that Ĥ operates on a function
and φ is an eigenfunction.
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For a one-dimensional problem, the eigenvalues can be calculated from
�
¯ Ĥφ(x)dx
φ(x)
Eφ = �
φ̄(x)φ(x)dx
Suppose that we do not know the eigenfunctions φ0 (x), φ1 (x), . . . , φn (x), we may still wish to
find an approximate way to calculate the observable energies, E1 , E2 , . . . , EN .
One method is to approximate the φ with a series of functions that match the boundary
conditions. For an electron in a one-dimensional box of length L, we could approximate φ with
φ = c1 x(L − x) + c2 [x(L − x)]2 + . . . + cM [x(L − x)]M
=
M
�
ci fi
i=1
where fi ≡ xi (L − x)i .
Therefore, we could consider the energy to be a function of the approximating parameters:
�L
ˆ 1 f1 + c2 f2 + . . . + f3 )dx
(c1 f1 + c2 f2 + . . . + cM fM )H(c
E(c1 , c2 , . . . , cM ) = �0 L
(c1 f1 + c2 f2 + . . . + cM fM )(c1 f1 + c2 f2 + . . . + f3 )dx
0
�M �M
i=1
j=1 ci cj Hij
= �M �M
i=1
j=1 ci cj Fij
�
�
ˆ j dx and Fij ≡ fi fj dx.
where Hij ≡ fi Hf
For the following, use the free-electron Hamiltonian operator.
1. Show that Hij = Hji (which is a property of a Hermitian operator Ĥ).
2. Find the lowest energy—which will be the ground-state approximation—by using M = 2
and minimizing E(c1 , c2 ) with respect to c1 and c2 . Compare your results with the exact
ground-state energy E0 .
3. Show that your results above can be written as the determinant:
�
�
H11 − EF11 H12 − EF12
det
=0
H12 − EF12 H22 − EF22
4. How does the larger root of the above equation compare with the energy E1 for an electron
in a box?
5. Generalize your result to calculate the ground-state energy using approximation of arbitrary
accuracy M. Calculate and plot the ground-state approximation as a function of M.
Hint: Write functions that takes two functions as arguments and return a number.
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